Systems and methods for determinization and minimization a finite state transducer for speech recognition
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 G10—MUSICAL INSTRUMENTS; ACOUSTICS
 G10L—SPEECH ANALYSIS OR SYNTHESIS; SPEECH RECOGNITION; SPEECH OR VOICE PROCESSING; SPEECH OR AUDIO CODING OR DECODING
 G10L15/00—Speech recognition
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 G10L15/18—Speech classification or search using natural language modelling
 G10L15/183—Speech classification or search using natural language modelling using context dependencies, e.g. language models
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Abstract
Description
This is a Continuation application Ser. No. 08/781,368 filed Jan. 21, 1997, now abandoned. The entire disclosure of the prior application is hereby incorporated by reference herein in its entirety.
This invention relates to pattern recognition systems. More particularly, the invention relates to determinization and minimization in natural language processing.
Finitestate machines have been used in many areas of computational linguistics. Their use can be justified by both linguistic and computational arguments. Linguistically, finite automata, which are directed graphs having states and transitions, are convenient since they allow one to describe easily most of the relevant local phenomena encountered in the empirical study of language. They often provide a compact representation of lexical rules or idioms and clichés which appears as natural to linguists. Graphic tools also allow one to visualize and modify automata. This helps in correcting and completing a grammar. Other more general phenomena such as parsing contextfree grammars can also be dealt with using finitestate machines. Moreover, the underlying mechanisms in most of the methods used in parsing are related to automata.
From the computational point of view, the use of finitestate machines is mainly motivated by considerations of time and space efficiency. Time efficiency is usually achieved by using deterministic algorithms that reduce redundancy in the automata. The output of deterministic machines depends, in general linearly, only on the input size and can therefore be considered as optimal from this point of view. Space efficiency is achieved with classical minimization algorithms that collapse some nodes in the determinized automata in a reverse determinizing manner. Applications such as compiler construction have shown deterministic finitestate automata to be very efficient in practice. Finitestate automata also constitute now a rich chapter of theoretical computer science.
Their recent applications in natural language processing which range from the construction of lexical analyzers and the compilation of morphological and phonological rules to speech processing show the practical efficiency of finitestate machines in many areas.
Both time and space concerns are important when dealing with language. Indeed, one of the trends which clearly come out of new studies of language is a large increase in the size of data. Lexical approaches have shown to be the most appropriate in many areas of computational linguistics ranging from largescale dictionaries in morphology to large lexical grammars in syntax. The effect of the size increase on time and space efficiency is probably the main computational problem one needs to face in language processing.
The use of finitestate machines in natural language processing is certainly not new. Indeed, sequential finitestate transducers are now used in all areas of computational linguistics. Limitations of the corresponding techniques, however, are very often pointed out more than their advantages. This might be explained by the lack of description of recent works in computer science manuals.
It is therefore an object of this invention to provide a pattern recognition system that applies determinization means to achieve time efficiency and minimization means to achieve space efficiency.
It is a more particular object of this invention to provide the speech recognition system that allows pruning of the automata by determinization without loss of information and better control of the result while taking account the word content represented by weights applied to automata.
It is another object of this invention to provide the speech recognition system that also allows pruning of the automata by minimization without loss of information while using equivalent functions as determinization.
It is yet another object of this invention to provide the speech recognition system that further allows testing of given automata to determine whether it can be determinized by a determinizability test.
These and other objects of the invention are accomplished in accordance with the principles of the present invention by providing a system and method for optimal reduction of redundancy and size of a weighted and labeled graph. The system and method of the present invention presents labeling the graph by assigning contents to arcs in the graph, evaluating weights of each path between states by examining relationship between the labels, assigning the weight to each arc and determinizing the graph by removing redundant arcs with the same weights. The system and method may further provide the step of minimizing the graph by collapsing a portion of the states in the graph in a reverse determinizing manner.
In another aspect of the invention, the system and method of the present invention may be applied for use in a language processing. Such system and method presents receiving speech signals, converting the speech signals into word sequences, evaluating the probability that each word sequence would be spoken, interpreting the word sequences in terms of the graph having its arcs labeled with the word sequences and its path weighted with the evaluated probabilities and determinizing the graph by removing redundant word sequences in the graph during the interpretation. The system and method may further present minimizing the graph by collapsing a portion of the states in a reverse determirnizing manner during the interpretation.
Further features of the present invention, its nature and various advantages will be more apparent from the accompanying drawings and the following detailed description of the preferred embodiments.
FIG. 1 is a simplified block diagram of illustrative apparatus which can be operated in accordance with this invention.
FIGS. 2ab (collectively referred to as FIG. 2) are a flow chart of steps for carrying out an illustrative embodiment of the methods of this invention.
FIG. 3 is a graph of illustrative subsequential string to weight transducer.
FIG. 4 is an algorithm for determinization of a transducer γ_{1 }representing a power series on the semiring.
FIG. 5 is a graph of illustrative transducer μ_{1 }representing a power series defined on {R_{+}∪{∞}, min, +}.
FIG. 6 is a graph of illustrative transducer μ_{2 }obtained by power series determinization of μ_{1}.
FIGS. 79 are graphs of illustrative minimization process where β_{1 }represents a subsequential string to weight transducer, γ_{1 }represents a transducer obtained from β_{1 }by extraction and δ_{1 }represents a minimal transducer obtained from γ_{2 }by automata minimization.
FIG. 10 is a graph of illustrative subsequential transducer X that is not bisubsequential.
FIG. 11 is a graph of illustrative transducer β_{2 }obtained by reversing β_{1}.
FIG. 12 is a graph of illustrative transducer β_{3 }obtained by determinization of β_{2}.
FIG. 13 is a graph of illustrative transducer β_{4 }obtained by minimization of β_{3}.
FIG. 14 is a graph of illustrative word lattice W_{1 }for the utterance “which flights leave Detroit and arrive at SaintPetersburg around nine a.m.?”
FIG. 15 is a graph of equivalent word lattice W_{2 }obtained by determinization of W_{1}.
FIG. 16 is a graph of equivalent word lattice W_{3 }obtained by minimization from W_{2}.
FIG. 17 is a table of illustrative word lattices in the ATIS task.
FIG. 18 is a table of illustrative deterministic word lattices in the NAB task.
The following detailed description presents the description of a system and method for recognizing speech signals, theoretical basis, and algorithmatic tools employed to analyze the complexity of language. It will, however, be apparent to those skilled in the art that the methodology of the invention is broadly applicable to a variety of other pattern recognition systems for optimal reduction of redundancy and size of a weighted and label graph.
1. Overview
In the illustrative embodiment of speech recognition system shown in FIG. 1, speech signals are fed into microphone/acoustic analyzer 100, where it is analyzed into acoustic features. The speech signals are then passed to decoder 102 which requests a state and its outgoing transitions. Network cache 110 is first examined to see if that information has already been computed and stored in its memory (such memory being initially empty).
If so, that information from network cache 110 is returned to decoder 102. Otherwise, network cache 110 passes the request onto contextdependency transducer 112. Thereafter, decoder 102 analyzes the speech signals by utilizingthe following transducers: contextdependency transducer 112, lexicon/pronunciation transducer 114 and grammar transducer 116.
Word lattice processor 104 then requests the transitions or paths leaving each state or node to formulate a graph and explores the paths and prunes out the unpromising ones (based on their acoustic and grammar match scores). This process is repeated until the final network states are reached. Best/nbest transcriber 106 then chooses the best scoring path and display device 108 displays the outcome of the recognition hypothesis.
Some components of the speech recognition system in FIG. 1 may be a microprocessor or digital signal processor (“DSP”) hardware, such as the AT&T DSP 16 or DSP 32C, readonly memory “ROM” for storing software performing the operations discussed below, and random access memory (“RAM”) for storing results. Very large scale integration (“VLSI”) hardware embodiments, as well as a custom VLSI circuity in combination with a general purpose DSP circuit, may also be provided.
In FIG. 2, a flow chart illustrating the operations of the speech recognition system of FIG. 1 is depicted. In this illustrative example, the flow chart begins at step 200. Microphone/acoustic analyzer 100 detects audio signals at step 202 and captures their acoustic measurements at step 204. Decoder 102 (FIG. 1) receives this acoustic observation and generates contextdependency units in accordance with the appropriate context environment. For example, the contextdependency units are determined to fit naturally between their previous phone and their subsequent phone.
Contextdependency transducer 112 (FIG. 1) receives these contextdependency units from decoder 110 to generate phonetic units at step 208. Lexicon/pronunciation transducer 114 (FIG. 1) subsequently receives these phonetic units from contextdependency transducer 112 and generates words at step 210. Grammar transducer 116 (FIG. 1) finally receives these words from lexicon/pronunciation transducer 110 and generates word sequences or sentences at step 212.
Decoder 102 (FIG. 1) and word lattice processor 104 (FIG. 1) in conjunction interpret these sentences and formulate a lattice which is a weighted and directed graph at step 214. Word lattice processor 104 (FIG. 1) assigns the estimated probability that word sequences would be spoken in constructing the paths in the lattice. The lattice, thus, have nodes labeled with the word sequences and paths weighted with the probabilities. For example, the word lattice processor 104 (FIG. 1) may be connected to a database containing the probabilities of word sequences that were used in a particular literature such as Wall Street Journal so that the speech recognition system would be more effective in recognizing the speech of economists or financiers.
Word lattice processor 104 (FIG. 1) then examines the lattice to test determinizability at step 216. The determinizability determines whether the lattice is suitable for certain processing such as determinization or minimization to efficiently reshape the lattice.
If the lattice passes the determinizability test, word lattice processor 1041 (FIG. 1) applies determinization in step 218 which removes redundant paths from the lattice. The theoretical basis and related algorithms for determinization will be discussed in a greater detail herein below.
Word lattice processor 104 (FIG. 1) may further apply minimization in step 220 so as to collapse lattice nodes that have the same possible combination. The theoretical basis and related algorithms for minimization will also be discussed in a greater detail herein below.
Best/nbest transcriber 106 (FIG. 1) thereafter may extract the best possible transcription at step 220. Alternatively, best/nbest transcriber 106 may extract a number of best transcriptions and reduce them to one or n transcription(s).
Display device 108 (FIG. 1) finally displays the resulted transcriptions at step 224. A viewer, therefore, may see the text of the speech being automatically typed on a monitor of display device 108 (FIG. 1). The flow chart ends in step 226.
Next several sections provide definition and algorithms that were used in determinization step 218, determinizability step 216 and minimization step 218. Thereafter, there is description of practical application of these algorithms.
2. Definition
In the following, string to weight transducers are introduced for their use in natural language processing. However, it would be apparent to those skilled in the art that the methodology of this invention is broadly applicable to a variety of other transducers including string to string transducers.
This section also considers subsequential power series in the tropical semiring that includes the functions associated with subsequential string to weight transducers. Most power series defined on the tropical semiring considered in practice are subsequential.
The string to weight transducers are used in various domains such as language modeling, representation of a word or phonetic lattice, etc. They are used in the following way: one reads and follows a path corresponding to a given input string and outputs a number obtained by combining the weight outputs along this path. In most applications of natural language processing, the weights are simply added along a path, since output label weights are interpreted as negative logarithms of probabilities. In case the transducer is not deterministic or sequential, one proceeds in the same way for all the paths corresponding to the input: string and then collect all numbers obtained this way. In natural language processing and especially in speech processing, only the minimum of these numbers is kept as being relevant information. We are interested here in the use of such transducers in many applications to natural language processing involving a Hidden Markov Model (“HMM”) that represents possible sequences of acoustic observations in realizations of the elementary speech units such as phones. In all such applications, one searches for the best path. This is related to the Viterbi approximation often used in speech recognition.
Specifically, the use of an efficient string to weight transducers called subsequential transducer is examined. Subsequential string to weight transducers are also called weighted automata, or weighted acceptors, or probabilistic automata, or distance automata. Not all transducers are subsequential but we define algorithms which allow one to determinize these transducers whenever possible. More precisely, we shall say that a string to weight transducer is subsequential when it is deterministic and when it is provided with a final weight function, and an initial weight. These weights are added to those encountered along a path corresponding to a given string.
Referring to FIG. 3, for instance, when used with the input string “ab.,” the transducer outputs: 5+1+2+3=11, 5 being the initial and 3 being the final weight. More formally a subsequential transducer τ=(V, i, F, A, δ, σ, λ, ρ) is an 8tuple, where V is the set of its states, i is the initial state of VF is the set of final states of V, A is the input alphabet of V, δ is the state transition function which maps V×A to V, σ is the output which maps V×A to _{+}, λ is the initial weight and ρ is the final weight function mapping F to _{+}. Unless otherwise specified, the output weights are positive real numbers.
String to weight transducers clearly realize functions mapping A* to _{+}. A* is a sequential or psubsequential function, i.e., a function which can be represented by a sequential or psubsequential transducer. Since the operations we need to consider are addition and min, and since _{+} ∪{∞}, min, +) is a semiring, we can call these functions formal power series according to the terminology used in formal languages theory. A semiring is essentially a ring which lacks negation, that is in which the first operation does not necessarily admit inverse. , +, ·), and (N∪{∞}, min, +) are other examples of semiring. We shall also adopt the notations used in this field. The image by a formal power series S of a string w is denoted by (S, w) and is also called the coefficient of w in S. The notation S=Σ(S, w) w is often used to define a power series by its coefficients. The support of S, supp(S), is the language defined by
The fundamental theorem of Schützenberger, analogous of the Kleene's theorem for formal languages states that a formal power series S is rational if it is recognizable, that is realizable by a transducer with initial weights and final weights. The semiring _{+} ∪{∞}, min, +) is often distinguished from other semirings since it is involved in many optimization problems. It is called the tropical semiring which is the terminology used in principle to denote the semiring of natural integers (N∪{∞}, min, +) involved in optimization problems. Thus the functions we consider here are more precisely rational power series defined on the tropical semiring.
3. Determinization
This section introduces a theorem helping to characterize subsequential power series independently from the transducer representing them. It includes an algorithm to determinize a transducer representing a power series admitting this operation.
A distance d(u,v) on the string A* can be defined by:
where we denote by uΛv the longest common prefix of two words u and v in A*, and s denotes the length of the string s. The definition we gave for subsequential power series depends on the notion of transducers representing them. The following theorem gives a characterization of subsequential power series which is independent of a transducer realizing them. This is an extension of the characterization theorem of Choffrut for string to string functions.
Let S be a formal rational series defined on the semiring (_{+}∪{∞}, min, +), S is subsequential if and only if it has bounded variation. For example, assume that S is subsequential. Let τ=(V, i, F, A, δ, σ, λ, ρ) be a subsequential transducer. δ denotes the transition function associated with τ, σ denotes its output function, λ and ρ denote the initial and final weight functions, q and q′ are states of V, and a is an input symbol of the alphabet A. Let L be the maximum of the lengths of all output labels of τ:
and R the upper bound of all output differences at final states:
and define M as M=L+R. Let (u_{1}, u_{2}) be words, or more generally, strings, in (A*)^{2}. By definition of the distance d, there exists u ε A* that is a common perfix string to the strings u_{1 }and u_{2 }such that:
where v_{1 }and v_{2 }are the remaining portions of u_{1 }and u_{2}, respectively, after the common prefix string u, and d(u_{1}, u_{2}) is the distance from u_{1 }and u_{2}.
Hence,
Since
and
we have
Notice that if u_{1}≠u_{2}, R≦R·d(u1, u2). Thus
Therefore:
This proves that S is MLipschitzian and a fortiori that it has bounded variation. This also implies in particular that the subsequential power series over the tropical semiring define continuous functions for the topology induced by the distance d on A*.
Conversely, suppose that S has bounded variation. Since S is rational, it is recognizable and therefore there exists a transducer τ_{1}=(V_{1}, i_{1}, F_{1}, A, δ_{1}, σ_{1}, λ_{1}, ρ_{1}) recognizing S. Here, τ_{1 }might be nondeterministic. Therefore, the transition function δ_{1 }maps V_{1}×A to 2^{v1}, and the output function σ_{1 }maps V_{1}×A×V_{1 }to _{+}. The output of a transition from a state q to a next state q′ labeled with a, where a is a symbol of the alphabet A, is therefore denoted by σ_{1 }(q, a, q′). Also λ_{1 }is a function since τ_{1 }might have several initial states. We shall assume that τ_{1 }is trim and it has no nonaccessible state or noncoaccessible state. Nonaccessible states are those unreachable from initial states and noncoaccessible states those which admit no path to a final state. Any rational function can be represented by a trim transducer simply by removing these useless states from a transducer representing it.
The following is an algorithm to construct from the nondeterministic transducer τ_{1 }representing S, an equivalent deterministic transducer τ_{2}=(V_{2}, i_{2}, F_{2}, A, δ_{2}, σ_{2}, λ_{1}, ρ_{2},). The algorithm extends to the case of transducers which output weights our determinization algorithm for string to string transducers representing psubsequential functions as described in an article by one of the inventors (Mohri, “On some applications of finitestate automata theory to natural language processing,” Technical Report ICM 9422, Institut Gespard Monge, 1994) which is incorporated herein by reference.
FIG. 4 presents this algorithm. It describes the algorithm in the general case of a semiring (S,⊕,⊙,0,1) on which the transducer τ_{1 }is defined, where ⊕ is a summining operation, ⊙ is a multiplying operation, “0” is the identity value for the summing operation and “1” is the identity value for the multiplying operation. Indeed, the algorithm described here applies as well to transducers representing power series defined on many other semirings. In particular, it can be verified that (A*∪{∞},Λ,·,∞,ε), where Λ denotes the longest common prefix operation, · denotes the concatenation, ε is the empty set, and ∞ denotes a new element such that for any string w ε (A*∪{∞}), wΛ∞=∞Λw=w and w·∞=∞·w=∞, also defines a left semiring. Considering this semiring in the algorithm of FIG. 4 gives the determinization for subsequentiable string to string transducers. The algorithm also applies to other semirings such as (_{+}+, ·) for instance. Since the cross product of two semirings also defines a semiring, this algorithm also applies to the cross product of (A*∪{∞},Λ,·,∞,ε) and (_{+}∪{∞}, min, +). This allows one to directly determinize transducers with both weight and string outputs admitting determinization. Thus, for the purpose of the algorithm we consider here, one can replace ⊕ by min. and ⊙ by + in the algorithm of FIG. 4. That is λ_{2} ^{−1 }should also be interpreted as −λ, and [σ_{2}(q_{2},a)]^{−1 }as −σ_{2}(q_{2}, a). The sets, Γ(q_{2},a), γ(q_{2},a) and ν(q_{2},a) and γ(q_{2},a) used in the algorithm are defined by:
The algorithm is similar to the powerset construction used for the determinization of nondeterministic automata. However, since the output of two transitions bearing the same input label might differ, here one can only output the minimum of these outputs in the resulting transducer and therefore needs to keep track of the remaining weight.
Subsets are thus formed of pairs of state and weight. Notice that several transitions might reach here the same state with a priori different remaining weights. Since one is only interested in the best path corresponding to the minimum weight, one can keep the minimum of these possible distinct weights for a given state element of a subset (line 11 of the algorithm of FIG. 4).
The algorithm is also illustrated by FIGS. 56. Notice that the input ab admits several outputs in u_{1}: {1+1=2, 1+3=4, 3+3=6, 3+5=8}. Only one of these outputs (2, the smallest one) is kept in the determinized transducer μ_{2 }since in the tropical semiring one is only interested in the minimum outputs for any given string. Most power series defined on the tropical semiring considered in practice are subsequential.
FIG. 5 shows a nondetermiaistic finitestate transducer u_{1}. As shown in FIG. 5, the nondeterministic finitestate transducer ul has four states, a start state 0, an intermediate state 2, and final states 1 and 3. In particular, given an input “a” received at the start state 0, the nondeterministic finitestate transducer u_{1 }can move from the start state 0 to the final state 1 with weight 3 or to the intermediate state 2 with weight 1. Similarly, if an input “b” is received at the start state 0, the nondeterministic finitestate transducer ul will move from the start state to the final state 1 with weight 1 or to the intermediate state 2 with weight 4. Similarly, at the final state 1, when the input “b” is received, the nondeterministic finitestate transducer u_{1 }will move from the final state 1 to the final state 3 with a weight of either 3 or 5. Likewise, at the intermediate state 2, when the input “b” is received, the nondeteiministic finitestate transducer u_{1 }will move from the intermediate state 2 to the final state 3 with a weight of either 1 or 3. Thus, the finitestate transducer u_{1 }is nondeterministic, because at least one state, and in this case, at states 0, 1, and 2, for a given input, there is more than one path which can be taken from that state to either different states or on different paths to the same state.
FIG. 4 is a pseudocode outlining the determinization process, where τ_{1 }is the input nondeterministic finitestate transducer and τ_{2 }is the output deterministic finitestate transducer. The nondeterministic finitestate transducer u_{1 }shown in FIG. 5 is one example of such a nondeterministic finitestate transducer τ_{1}, while the deterministic finitestate transducer u_{2 }shown in FIG. 6 is one example of such a nondeterministic finitestate transducer τ_{2}.
In particular, starting in step 1, the final state set F_{2 }of the deterministic finitestate transducer τ_{2 }to be generated is set to the empty set. Then, in steps 2 and 3, the set of initial states I_{1 }of the nondeterministic finitestate transducer τ_{1 }and the minimum weight λ_{2 }of weights λ_{1}, of the initial states i are used to create a single initial state i_{2 }of the deterministic finitestate transducer τ_{2}. As shown in the corresponding initial state 0 of FIG. 6, this single initial state i_{2 }will have one (or more) substates, each indicating the original state i of the nondeterministic finitestate transducer τ_{1 }and the difference between the weight of that original state i and the minimum weight λ_{2}.
Next, in step 4, the single initial state i_{2 }of the deterministic finitestate transducer τ_{2 }is pushed on top of a queue Q. At this point, the queue Q contains only the single initial state i_{2 }of the deterministic finitestate transducer τ_{2}. In steps 5 and 6, while the queue Q is not empty, the top state on the queue Q is popped as the current state q_{2 }of the deterministic finitestate transducer τ_{2}. Next, in state 7, the current state q_{2 }is checked to determine if it contains a substate (q, x) that was a final state of the nondeterministic finitestate transducer τ_{1}. If so, in step 8, the current state q_{2 }is added to the set of final states F_{2 }of the deterministic finitestate transducer τ_{2}. Then, in step 9, the final output weight ρ associated with the current state q_{2 }is the minimum of the sum, for each substate q of the current state q_{2}, of the remainder x of that substate q and the final output weight p associated with that substate q.
In step 10, the set of transition labels a of all transitions from all substates of the current state is determined. For example, when the current state is state 0 of FIG. 5, the set of transition labels a includes “a, b”. Similarly, when the current state is state 1 or state 2, the set of transition labels a includes “b∞. In step 10, each transition label a of the set of transition labels is selected in turn as the current label. Next, in steps 11 and 12, a new possible state of the deterministic finitestate transducer τ_{2 }is created based on the states of the nondeterministic finitestate transducer τ_{1 }reachable from all of the substates of the current state q_{2 }based on the current transition label. Thus, when 0 is the current state and “a” is the current transition label, the new possible state includes the substates 1 and 2. Similarly, when the current state is the state 1′ of FIG. 6, the new state includes only the final state 3. In addition, creating the new possible state includes generating a new transition (δ_{2}(q,a)) from the current state q_{2 }to the new state, determining the weight (σ_{2}(q,a)) of that new transition and determining the remainders x associated with each of the substates q of the possible new state.
In step 13, the new possible state is checked to determine if it is identical to some previously determinized state in the deterministic finitestate transducer τ_{2 }being created. That is, in step 13, the new state is checked to determine if it is the same state as some old state. To be the same state, the transitions and the substates q of the new state must be identical to the transitions and the substates q of the old state, and the remainders on each of the substates q must be same as the remainders on each of the corresponding substates q of the old state. In step 14, if the new state is truly new, in that it is not identical to some old state, the new state is pushed onto the queue Q.
Steps 1014 are repeated for each transition label a until the set of transition labels is empty. If the set of transitions is empty, then all of the transitions from all of the substates q of the current state q_{2 }have been determinized and no further analysis of the current state q_{2 }is necessary. In step 15 the current state q_{2 }is removed from the queue Q. Steps 615 are then repeated until the queue Q is empty.
In steps 11 and 12, a possible new state is created based on the substates q, if any, of the current state q_{2 }using the current transition label a. In step I1, thf weight σ_{2}(q_{2},a) of the current transition label a is determined as the minimum, for each transition between a substate q of the current state q_{2 }and a next state q′ having the current transition laibel a, of the weight σ_{1}, of that transition plus the remainder x of the current substate q.
Applying the determinization method outlined in FIG. 4 to, the nondeterministic finitestate transducer u_{1 }shown in FIG. 5 generates the deterministic finitestate transducer u_{2 }shown in FIG. 6. First, the start state 0′ of the deterministic finitestate transducer u_{2 }is pushed onto a stack queue Q. While the queue Q is not empty, the top state of the queue Q is popped from the queue Q. From the initial state 0 of the nondeterministic finitestate transducer u, shown in FIG. 5, the states 1 and 2 can be reached using the input symbol “a”. Thus, a new state 1′=[(1,x), (2,x)] reachable from the current state 0′ (=[(0,0)]) of the deterministic finitestate transducer u_{2 }with the input symbol “a” is established with a new transition t_{1}′,a,ρ,1′]. The weight ρ for the new transition t_{1}′ is the minimum weight of the transitions labeled “a” from the start state 0 to either of the reachable states 1 or 2, plus any remainder that may be assigned to a corresponding substate of the current state 0′. Because the current state 0′ is a start state, there is no remainder. Similarly, because the nondeterministic finitestate transducer u_{1 }shown in FIG. 5 has only a single start state, there is only one substate of the current state 0′.
As shown in FIG. 5, the transition labeled “a” from the start state 0 to the state 2 has the minimum weight, 1. Thus, the weight ρ for the transition t_{1}′ is 1. Because the weight assigned to the new transition t_{1}′ does not capture the full weight of the other alabeled transition from the start state 0 to the state 1, each of the substates 1 and 2 of the new state 1′ are assigned remainders x. In particular, the remainder for the substate 1 is 2, while the remainder for the substate 2 is zero. Thus, the new state 1′ is rewritten as [(1,2),(2,0)].
Additionally, because state 1 is a final state, any state of the deterministic finitestate transducer u_{2 }will be a final state, and will have an output value. As indicated above, the minimum output weight for an input of “ab” is 2. This weight is output from state 1′, as shown in FIG. 6.
Similarly, from the start state 0 in the original nondeterministic finitestate transducer u_{1 }shown in FIG. 5, states 1 and 2 can also be reached from the current state 0′ using the input symbol “b”. Another new state 2′ is formed, and a new transition t._{2 }is formed from the current state 0′ to the new state 2′. The new transition t_{2}′ is again assigned a weight which is equal to the minimum weight of the blabeled transitions from the current state 0′ to the substates of the new state 2′. Likewise, each of the substates of the new state 2′ are assigned remainder weights. This is also shown in FIG. 6.
Because each of the new states 1′ and 2′ is not identical to any other previously determinized, or old, state, each of the new states 1′ and 2′ are added to the queue Q. Because all of the transitions from the current state 0′ have now been analyzed and determinized, the new state 1′ is popped from the queue Q. In particular, the current state 1′ has two substates 1 and 2. Each of these substates 1 and 2 in the original nondeterministic finitestate transducer u_{1 }shown in FIG. 5 has multiple exiting transitions, but each such transition is labeled with the same label “b”. Furthermore, each of these transitions moves to the final state 3. Thus, in the nondeterministic finitestate transducer u_{1 }shown in FIG. 5, the final state 3 can be reached from both the substates 1 and 2 of the current state 1′ using the input symbol “be”. In addition, there are two such blabeled transitions from each of the substates 1 and 2. For each such original transition, a transition weight ρ_{t }for each transition is determined as the sum of the weight assigned to that transition and the remainder associated with the original substate corresponding to that transition in the current state 1′. The weight to be associated with the new transition from the current state 1′ to the new state 3′ is then the minimum of the transition weights ρ_{t}. In particular, the minimum transition weight ρ_{r }for the new state 3′ is 1. Since there is only one reachable state for the input table “b” for the current state 1′, the minimum weight p for the blabeled transition for the current state 1′ is 1.
The minimum transition weight ρ is then used as the weight for a new transition t_{3}′ from the current state 1′ to the new state 3′. Because there is only one reachable state, the final state 3, reachable in the original nondeterministic finitestate transducer u_{1 }shown in FIG. 5 from the current state 1′, the new state established from the current state 1′ reachable using the input symbol b is 3′=[(3,x)]. Using the same procedure, a transition t_{4}′from the state 2′ to the state 3′ can be formed having a weight of 3. This is shown in FIG. 6. The remainder on the new state 3′ is the minimum remainder resulting from the remainders obtained by adding the remainders in the states 1′ and 2′ and the weights on the corresponding original transitions in the nondeterministic finitestate transducer u_{1}, less the weight on the corresponding new transitions t_{3}′ and t_{4}′. Accordingly, this new state 3′ thus has a remainder of zero, resulting in the complete new state 3′=[(3,0)]. The end result is thus shown in FIG. 6.
Generally when the deserminization algorithm terminates, the resulting transducer τ_{2 }is equivaent to r_{1}. For example, if θ_{1}(q,w,q′) denotes the minimum susof the outputs of all paths from q to q′ by construction, line 2 of FIG. 4 becomes:
Further, if the residual output associated to q in the subset δ_{2}(i_{2},w) is defined as the weight c(q,w) associated to the pair containing q in δ_{2}(i_{2},w), it is not hard to show by induction on w that the subsets contructed by the algorithm are the sets δ_{2}(i_{2},w), w εA*, such that:
Notice that the size of a subset never exceeds V: card (δ_{2}(i_{2},w))≦V_{1}. A state q belongs at most to one pair of a subset since for all paths reaching q, only the minimum of the residual outputs is kept. Notice also that, by definition of min, in any subset there exists at least on state q with a residual output c(q,w) equal to 0.
A string w is accepted by τ_{1 }if and only if there exists qεF_{1 }such that qε(δ_{1}(i_{1}, w). Using the equations set above, it is accepted if and only if δ_{2 }(i_{2}, w) contains a pair (q,c(q,w)) with qεF_{1}. This is exactly the definition of the final states F_{2 }in line 7 of FIG. 4. So τ_{1 }and τ_{2 }accept the same set of strings.
Let wεA* be a string accepted by τ_{1 }and τ_{2}. The definition of ρ_{2 }in the algorithm of line 9 of FIG. 4 gives:
Thus, if we denote by S the power series realized by τ_{1}, we have:
Hence:
Let T=(Q,ε,I,F,E,λ,ρ) be a stringtoweight transducer, περ→^{w}q a path in T from the state pεQ to qεQ, and π′εp′→^{w}q′ a path from p′εQ to q′εQ both labeled with the input string wεΣ*. Assume that the lengths of π and π′ are greater than Q^{2}−1, then there exist strings u_{1}, u_{2}, u_{3 }in Σ*, and states p_{1}, p_{2}, p′_{1}, and p′_{2 }such that u_{2}>0, u_{1}u_{2}u_{3}=w and such that π and π′ be factored in the following way:
Given two transducers T_{1}=(Q_{1},Σ, I_{1},F_{1},E_{1},λ_{1},ρ_{1}) and T_{2}=(Q_{2},Σ,I_{2},F_{2},E_{2},λ_{2}, ρ_{2}), the cross product of T_{1 }and T_{2 }may be defined as the transducer:
with outputs in R_{+}×R_{+} such that t=((q_{1},q_{2}),a, (x_{1},x_{2}), (q′_{1},q′_{2})) ε Q_{1}×Σ×R_{+}×R_{+}×Q_{2 }is a transition of T_{1}×T_{2}, namely tεE, if and only if (q_{1}, a, x_{1}, q′_{1}) ε E_{1 }and (q_{2}, a, x_{2}, q′_{2}) εE_{2}. λ and ρ can be also defined by: ∀(i_{1}, i_{2}) ε I_{1}×I_{2}, λ(i_{1}, i_{2})=(λ(i_{1}), λ_{2}(i_{2})), ∀(f_{1}, f_{2}) ε F_{1}×F_{2}, ρ(f_{1}, f_{2})=(ρ_{1}(f_{1}), ρ_{2}(f_{2})).
Consider the cross product of T with itself, T×T. Let π and π′ be two paths in T with lengths greater than Q^{2}−1, (M>Q^{2}−1):
then:
is a path in T×T with length greater than Q^{2}−1. Since T×T has exactly Q^{2 }states, π admits at least one cycle at some state (p_{1},p′_{1},) labeled with a non empty input string u_{2}. This shows the existence of the factorization above.
Let , τ_{1}=(Q_{1},Σ,I_{1},F_{1},E_{1},λ_{1},ρ_{1}) be a stringtoweight transducer defined on the tropical semiring. If τ_{1 }has the twins property then it is determinizable. For example, assume that τ has the twins property. If the determinization algorithm does not halt, there exists at least one subset of 2^{Q}, {q_{0}, . . . , q_{m}}, such that the algorithm generates an infinite number of distinct weighted subsets {(q_{0}, c_{0}), . . . , (q_{m}, c_{m})}.
Then it is necessarily m>1. Indeed, as previously mentioned, in any subset there exists at least one state q_{i }with a residual output c_{i}=0. If the subset contains only one state q_{0}, then c_{0}=0. So there cannot be an infinite number of distinct subsets {(q_{0}, c_{0})}.
Let W⊂Σ* be the set of strings w such that the states of δ_{2}(i_{2}, w) be {q_{0}, . . . , q_{m}}. We have: ∀ωε A, δ_{2}(i_{2}, ω)={(q_{0}, c(q_{0}, ω)), . . . , (q_{m}, c(q_{m}, ω))}. Since W is infinite, and since in each weighted subset there exists a null residual output, there exist i_{0}, 0≦i_{0}≦m, such that c,(q_{i0}, ω)=0 for an infinite number of strings ω ε W. Without loss of generality, it can be assumed that i_{0}=0.
Let B⊂W be the infinite set of strings ω for which c(q_{0}, ω))=0. Since the number of subsets {(q_{0}c(q_{0}, ω)), . . . , (q_{m}, c(q_{m}, ω)) }, ωεB, is infinite, there exists j, 0<j≦m, such that c(q_{j}, ω) be distinct for an infinite number of strings ω ε B. Without loss of generality, j=1 can be assumed.
Let C⊂B be an infinite set of strings ω with c(q_{1}, ω) all distinct. (q_{0}, q_{1}) can be defined as the finite set of differences of the weights of paths leading to q_{0 }and q_{1 }labeled with the same string ω, ω≦Q_{1}^{2}−1:
It can be shown that {c(q_{1}, ω):ωεC}⊂R(q_{0}, q_{1}). This will yield a contradiction with the infinity of C, and will therefore prove that the algorithm terminates.
Let ωεC, and consider a shortest path π_{0 }from a state i_{0}ε I to q_{0 }labeled with the input string ω and with total cost σ (π_{0}). Similarly, consider a shortest path π_{1 }from i_{1 }ε I to q_{1 }labeled with the input string ω and with total cost σ (π_{1}). By definition of the subset construction we have: (λ(i_{1})+σ(π_{1}))−(λ(i_{0})+σ(π_{0}))=c(q_{1}, ω). Assuming that ω>Q_{1}^{2}−1, there exists a factorization of π_{0 }and π_{1 }of the type:
with u_{2}>0. Since p_{0 }and p_{1 }are twins, θ_{1}(p_{0}, u2, P_{0})=θ_{1}(p_{1}, u_{2}, p_{1}). π′_{0 }and π′_{1 }may be defined by:
Since π and π′ are shortest paths, σ(π_{0})=σ(π′_{0})+θ_{1}(p_{0}, u_{2}, p_{0}) and σ(π_{1})=σ(π′_{1})+θ_{1}(p_{1}, u_{2}, p_{1}). Therefore: (λ(i_{1})+σ(π′_{1})−(λ(i_{0})+σ(π′_{0})=c(q_{1}, ω). By induction on ω, the shortest paths π_{0 }and π_{1 }from i_{0 }to q_{0 }(resp. i_{1 }to q_{1}) can be found with length less or equal to Q_{1}^{2}−1 and such that λ(i_{1})+σ(π_{1})) (λ(i_{0})+σ(π_{0}))=c(q_{1}, ω). Since σ(π_{1})−σ(π_{0})εR(q_{0}, q_{1}), c(q_{1}, ω) εR(q_{0}, q_{1}), and C is finite.
There are transducers that do not have the twins property and that are still determinizable. To characterize such transducers one needs more complex conditions. Those conditions are not described in this application. However, in the case of trim unambiguous transducers, the twins property provides a characterization of determinizable transducers.
Let τ_{1}=(Q_{1},Σ,I_{1},F_{1},E_{1},λ_{1},ρ_{1}) be a trim unambiguous stringtoweight transducer defined on the tropical semiring. Then τ_{1 }is determinizable if and only if it has the twins property.
For example, if τ_{1 }has the twins property, then it is determinizable. Assume now that τ does not have the twins property, then there exist at least two states q and q′ in Q that are not twins. There exists (u, v) εΣ* such that: ({q, q′} ⊂ δ_{1}(I, u), q ε δ(q, v) , q′εδ_{1}(q′, v)) and θ_{1}(q, v, q)≠θ_{1}(q′, v, q′). Consider the weighted subsets δ_{2}(i_{2}, uv^{k}) , with kεN, constructed by the determinization algorithm. A subset δ_{2}(i_{2}, uv^{k}) contains the pairs (q, c(q, uv^{k})) and (q′, c(q′, uv^{k})) . It can be shown that these subsets are all distinct. The determinization algorithm, therefore, does not terminate if τ_{1 }does not have the twins property.
Since τ_{1 }is a trim unambiguous transducer, there exists only one path in τ_{1 }from I to q or to q′ with input string u. Similarly, the cycles at q and q′ labeled with v are unique. Thus, there exist i ε I and i′ ε I such that:
Let λ and θ be defined by:
We have:
Since θ≠0, this equation shows that the subsets δ_{2}(i_{2}, uv^{k}) are all distinct.
Notice that the power series determinization algorithm is equivalent to the usual determinization of automata when all output labels are equal to 0, and all final weights equal to 0. The corresponding power series is then the characteristic function of its support supp(S) and represents exactly a regular language. The subsets considered in the algorithm are then exactly those obtained in the powerset determination of automata, the remainder output being constant and equal to 0.
Both space and time complexity of the determinization algorithm for automata is exponential. There are minimal deterministic automata with an exponential size with respect to nondeterministic ones representing them. A fortiori the complexity of the determinization algorithm in the weighted case we just described is also exponential. However, in some cases in which the degree of nondeterminism of their initial transducer is high, the determinization algorithm appears as fast and the resulting transducer has fewer number of states than the initial one. Such cases which appear in speech recognition are discussed here. Also presented is a minimization algorithm which allows one to reduce the size of subsequential transducers representing power series.
The complexity of the application of subsequential transducers is linear in the size of the string to which they apply. This property makes it interesting to use the power series determinization in order to speed up the application of transducers. However, since the algorithm only applies to subsequential power series, we need an algorithm to test this property. The following section is devoted to the definition of such an algorithm.
4. Test of determinizability
The characterization of determinizable transducers described in section 3 herein above leads to the definition of an algorithm for testing the determinizability of trim unambiguous transducers. Before describing the algorithm, it is shown that it suffices to examine a finite number of paths to test the twins property.
For example, let τ_{1}=(Q_{1}Σ,I_{1},F_{1},E_{1},λ_{1},ρ_{1}) be a trim unambiguous stringtoweight transducer defined on the tropical semiring. τ_{1 }has the twins property if and only if ∀(u,v) ε(Σ*)^{2}, uv≦2Q_{1}^{2}−1,
Clearly if τ_{1 }has the twins property, then the above equation holds. Conversely, if the above equation holds, then it also holds for any (u,v)ε(Σ*)^{2}, by induction on uv. Consider (u,v)ε(Σ*)^{2 }and (q, q′)εQ_{1}^{2 }such that:{q,q′}⊂δ_{1}(q,v),q′εδ_{1}(q′,v). Assume that uv>2Q_{1}^{2}−1 with v>0. Then either u>Q_{1}^{2}−1 or v>Q_{1}^{2}−1.
Assume that u>Q_{1}^{2}−1. Since τ_{1 }is a trim unambiguous transducer there exists a unique path π in τ_{1 }from iεI to q labeled with the input string u, and a unique path π′ from i′εI to q′. In view of above equation, there exist strings u_{1}, u_{2}, u_{3 }in Σ*, and states p_{1}, p_{2}, p′_{1}, and p′_{2 }such that u_{2}>0, u_{1}u_{2}u_{3}=u and such that π and π′ be factored in the following way:
Since u_{1}u_{2}v<uv, by induction θ_{1}(q,v,q)=θ_{1}(q′,v,q′).
Next, assume that V>Q_{1}^{2}−1. Then, there exist strings v_{1},v_{2},v_{3 }in Σ*, and states q_{1},q_{2},q′_{1}, and q′_{2 }such that v_{2}>0, v_{1}v_{2}v_{3}=v and such that π and π′ be factored in the following way:
Since uv_{1}v_{3}<uv, by induction, θ_{1}(q,v_{1}v_{3},q)=θ_{1}(q′,v_{1}v_{3},q′). Similarly, since uv_{1}v_{2}<uv, θ_{1}(q_{1},v_{2},q_{1})=θ_{1}(q′_{1}, v_{2}, q′_{1}) . τ_{1 }is a trim unambiguous transducer, so:
Thus, θ_{1}(q,v,q)=θ_{1}(q′,v,q′).
If τ_{1}=(Q_{1}, Σ,I_{1},F_{1},E_{1},λ_{1},ρ_{1}) is a trim unambiguous stringtoweight transducer defined on the tropical semiring, then there exists an algorithm to test the determinizability of τ_{1}.
For example, testing the determinizability of τ_{1 }is equivalent to testing for the twins property. An algorithm to test this property is defined, specifically for testing the sequentiability of stringtostring transducers. It is based on the construction of an atom A=(Q,I,F,E) similar to the cross product of τ_{1 }with itself.
Let k⊂_{+} be the finite set of real numbers defined by:
A can be defined by the following:
The set of states of A is defined by Q=Q_{1}×Q_{1}×K,
The set of initial states by I=I_{1}×I_{1}×{0}
The set of final states by F=F_{1}×F_{1}×K,
The set of transictions by:
By construction, two states q_{1 }and q_{2 }of Q can be reached by the same string u, u≦2Q_{1}^{2}−1, if and only if there exists cεK such that (q_{1},q_{2},c) can be reached from I in A. The set of such (q_{1},q_{2},c)is exactly the transitive closure of I in A. The transitive closure of I can be determined in time linear in the size of A,O(Q+E).
Two such states q_{1 }and q_{2 }are not twins if and only if there exists a path in A from (q_{1},q_{2},0) to (q_{1},q_{2},c), with c≠0. Indeed, this is exactly equivalent to the existence of cycles at q_{1 }and q_{2 }with the same input label and distinct output weights and it suffices to test the twins property for strings of length less than 2Q_{1}^{2}−1. So the following gives an algorithm to test the twins property of a transducer τ_{1}:
1. Compute the transitive closure of I: T(I) .
2. Determine the set of pairs (q_{1},q_{2}) of T(I) with distinct states q_{1}≠q_{2}.
3. For each such {q_{1},q_{2}} compute the transitive closure of (q_{1}, q_{2}, 0) in A. If it contains (q_{1},q_{2},c) with c≠0, then T_{1 }does not have the twins property.
The operations used in the algorithm (computation of the transitive closure, determination of the set of states) can all be done in polynomial time with respect to the size of A, using classical algorithms.
This provides an algorithm for testing the twins property of an unambiguous trim transducer T. It is very useful when one knows auto be unambiguous.
In many practical cases, the transducer one wishes to determine is ambiguous. It is always possible to construct an unambiguous transducer T′ from T. The complexity of such a construction is exponential in the worst case. Thus the overall complexity of the test of determinizability is also exponential in the worst case.
Notice that if one wishes to construct the result of the determinization of T for a given input string w, one does not need to expand the whole result. of the determinization, but only the necessary part of the determinized transducer. When restricted to a finite set the function realized by any transducer is subsequentiable since it has bounded variation. Acyclic transducers have the twins property, so they are determinizable. Therefore, it is always possible to expand the result of the determinization algorithm for a finite set of input strings, even if T is not determinizable.
5. Minimization
This section presents a minimization algorithm which applies to subsequential string to weight transducers. The algorithm is very efficient. The determinization algorithm can also be used in many cases to minimize a subsequential transducer.
A minimization algorithm for subsequential power series is defined on the tropical semiring which extends the algorithm defined by an article by one of the inventors (Mohri, “Minimization of sequential transducer,” Lecture Notes in Computer Science 4, 1994, Institut Gaspard Monge, Universit Marne la Vallee) in the case of string to string transducers. This article is incorporated herein by reference. An algorithm allowing one to minimize the representation of power series was defined by Schütetzenberger. However, Schützenberger's algorithm only applies when the semiring considered is a field. Thus, it does not apply to the tropical semiring. Our minimization algorithm applies to subsequential power series defined /_{+}, ·) other semirings, in particular to those defined on for which the algorithm by Schüttzenberger was defined.
For any subset L of a finite set A* of the alphabet A (or of a finite set Σ* of the alphabet Σ) and any string u we define u^{−1}L by:
Recall that L is a regular language if and only if there exists a finite number of distinct u^{−1}L. In a similar way, given a power series S we define a new power series u^{−1}S by:
One can prove that S is a rational power series if and “only if it admits a finite number of independents u^{−1}S. This is the equivalent for power series of the Nerode's theorem for regular languages.
For any subsequential power series S, we can now define the following relation on A*:
It is easy to show that R_{s }is an equivalence relation. u^{−1 }supp(S)=v^{−1 }supp(S)) defines the equivalence relation for regular languages. R_{s }is a finer relation. The additional condition in the definition of R_{s }is that the restriction of the power series [u^{−1}S−v^{1}S] to u^{−1}supp(S)=v^{−1}supp(S) is constant. The following shows that if there exists a subsequent transducer T computing S with a number of states equal to the number of equivalence classes of R_{s}, then T is a minimal transducer computing f.
For example if there is a subsequent power series defined on the tropical semiring, R_{s }has a finite number of equivalence classes. This number is bounded by the number of states of any subsequent transducer realizing S. For example, let T=(V, i, F, A, δ, σ, λ, ρ) be a subsequent transducer realizing S. Clearly,
Also, if u^{−1}supp(S)=v^{−1}supp(S), ∀(u, v)ε(A*)^{2},
The existence of a minimal subsequent transducer representing S may be shown. For any sequential function S, there exists a minimal subsequent transducer computing it. Its number of states is equal to the index of R_{s}. For instance, given a subsequent series S, a series f may be defined by:
A subsequent transducer T=(V, i, F, A,δσ,λ,ρ) may then defined by:
where we denote by {overscore (u)} the equivalence class of uεA*.
Since the index of R_{s }is finite, V and F are well defined. The definition of δ does not depend on the choice of the element u in {overscore (u)} since for any a εA, u R_{s }v implies (ua) R_{s }(va). The definition of τ is also independent of this choice since by definition of R_{s}, if uR_{s}v, then (ua) R_{s }(va) and there exists kε such that ∀wεA*, (S, uaw)−(S, vaw)=(S, uw)−(S, vw)=k. Notice that the definition of σ, implies that:
Thus, T realizes S:
Given a subsequential transducer T=(V, i, A, δ, σ, λ, ρ), each state qεV, d(q) can be defined by:
A new operation of pushing can be defined which applies to any transducer T. In particular, if T is subsequential and the result of the application of pushing to T is a new subsequential transducer T′=(V, i, F, A, δ, σ′, λ′, ρ′) which only differs from T by its output weights in the following way:
Since d(δ(q, a))−d(q)+σ(q,a)≦0, T′ is well defined.
Let T′ be the transducer obtained from T by pushing. T′ is a subsequential transducer which realizes the same function as T. Also,
Therefore, since δ(i, w)εF=>d(δ(i,w))=ρ(δ(iw)), λ′+σ−(i, w)+p′(δ(i, w))=λ+d(i)+σ(q, w)+ρ(δ(i, w))−d(i)+0
The following defines the minimization algorithm.
Let T be a subsequential transducer realizing a power series on the tropical semiring. Then applying the following two operations:
pushing
automata minimization leads to a minimal transducer.
The minimization step consists of considering pairs of input label and associated weights as a single label and of applying classical minimization algorithms for automata. In general, there are several distinct minimal subsequential transducers realizing the same function. However, their topology is identical. They only differ by their output labels which can be more or less delayed. The result of the application of pushing to any minimal transducer is however unique. It is the minimal transducer defined in the proof of the existence of a mimimal subsequential transducer computing the subsequential function S.
Notice that if a new super final state Φ to which each final state q is connected by a transition of weight p(q) is introduced, then d(q) in the definition of T′ is exactly the length of a shortest path from Φ to q. Thus, T′ can be obtained from T by the classical singlesource shortest paths algorithms such as that of Dijkstra if the transition weights are low. This algorithm can be extended to the case where weights are negative. If there is no negative cycle the BellmanFord algorithm can be used. In case the transducer is acyclic, a linear time algorithm based on a topological sort of the graph allows one to obtain d.
Once the function d is defined, the transformation of T into T′ can be done in linear time, namely O([V]+[E]), if we denote by E the set of transitions of T. The complexity of pushing is therefore linear (O([V]+[E])) if the transducer is acyclic, C([E]+[V]log [V]) in the general case if we use Fibonacci heaps in the implementation of the algorithm or ([E]+[V][W]) in case the maximum transition weight W is small using the algorithm in an article by Ahuja. Faster algorithm for the shortest path problems.
Also, in case the transducer is acyclic, one can use a specific minimization algorithm with linear. time complexity. Therefore, the overall complexity of the minimization algorithm for a subsequential transducer is 0 ([V]+[E]) in case T is acyclic and 0 ([E]log [V]) in the general case. Thus, its complexity is always as good as that of classical automata minimization.
FIGS. 7, 8 and 9 illustrate the minimization process. β_{1 }in FIG. 7 represents a subsequential string to weight transducer. Notice that the size of β_{1 }cannot be reduced using the automata minimization. γ_{1 }in FIG. 8 represents the transducer obtained by pushing, and δ_{1 }in FIG. 9 is a minimal transducer realizing the same function as β_{1 }in the tropical semiring. In particular, the minimal transducer δ_{1 }shown in FIG. 9 is obtained from the transducer γ_{1 }shown in FIG. 8 by collapsing states having identical output transitions, i.e., transitions leaving the states and with the same input label and weight, into a single state. As shown in FIGS. 8 and 9, the states 2 and 4 of the transducer γ_{1 }shown in FIG. 8 have single output transistions labeled “c” and having a weight of “0”. Thus, the states 2 and 4 of the transducer γ_{1 }shown in FIG. 8 are collapsed into the single state 2 of the transducer γ_{1 }shown in FIG. 9. Similarly the states 3 and 5 of the transducer γ_{1 }shown in FIG. 8 each have a first output transition labeled “d” and having a weight of “6” and a second output transition labeled “e” and having a weight of “0”. Thus, the states 3 and 5 of the transducer γ_{1 }shown in FIG. 8 are collapsed into the single state 3 of the transducer δ_{1 }shown in FIG. 9.
The transducer obtained by this algorithm and defined in the proof of the existence of a minimal subsequential transducer computting the subsequential function S has the minimal number of states. One can ask whether there exists a subsequential transducer with the minimal number of transitions and computing the same function as a given subsequential transducer T. The following theorem brings a specific answer to this question.
A minimal subsequential transducer has also the minimal number of transitions among all subsequential transducers realizing the same function. This generalizes the analogous theorem which holds in the case of automata. For example, let T be a subsequential transducer with the minimal number of transitions. Clearly, pushing does not change the number of transitions of T and the automaton minimization which consists in merging equivalent states reduces or does not change this number. Thus, the number of transitions of the minimal transducer of T as previously defined is less or equal to that of T.
Given two subsequential transducers, one might wish to test their equality. The following theorem addresses this question.
Let A be a nondeterministic automaton. Then the automaton A′=(Q′, i′, F′, Σ, δ′) obtained by reversing A, applying determinization, rereversing the obtained automaton and determinizing it is the minimal deterministic automaton equivalent to A.
A rational power series S can be stated as bisubsequential when S is subsequential and the power series S_{R}=Σ_{weΣ*}(S, w^{R}) w is also subsequential. However, not all subsequential transducers are bisubsequential. FIG. 10 shows a transducer representing a power series S that is not bisubsequential. S is such that:
The transducer of FIG. 10 is subsequential so S is subsequential. But the reverse S^{R }is not because it does not have bounded variation. Indeed, since:
We have:
One can give a characterization of bisubsequential power series defined on the tropical semiring in a way similar to that of stringtostring transducers. In particular, the previous sections shows that S is bisubsequential if and only if S and S^{R }have bounded variation.
Similarly bideterminizable transducers are defined as the transducers T also defined on the tropical semiring admitting two applications of determinization in the following way:
1. The reverse of T,T^{R }can be determinized. We denote by det(T^{R}) the resulting transducer.
2. The reverse of det(T^{R}), [det(T^{R})]^{R }can also be determinized. We denote by det([det(T^{R})]^{R}) the resulting transducer.
In this definition, it is assumed that the reverse operation is performed simply by reversing the direction of the transitions and exchanging initial and final states. Given this definition, we can present the extension of the use of determinization algorithm for power series as minimization tool for bideterminizable transducers.
If T be a bideterminizable transducer defined on the tropical semiting, then the transducer det ([det (T^{R})]^{R}) obtained by reversing T, applying determinization, rereversing the obtained transducer and determinizing it is a minimal subsequential transducer equivalent to T.
T can be denoted by:
T′=(Q′,i′,F′,Σ,δ′,σ′,λ′,ρ′) det([det(T ^{R})]^{R})
the transducer obtained from T by pushing.
The double reverse and determinization algorithms clearly do not change the function T realizes. So T′ is a subsequential transducer equivalent to T. Then it is necessary to show that T′ is minimal. This is equivalent to showing that T″ is minimal since T′ and T″ have the same number of states. T_{1 }is the result of a determinization, hence it is a trim subsequential transducer. It can be shown that T′=det (T_{1} ^{R}) is minimal if T_{1 }is a trim subsequential transducer. T is not required to be subsequential.
Let S_{1 }and S_{2 }be two states of T″ equivalent in the sense of automata. It can be shown that S_{1}=S_{2}, namely that no two distinct states of T″ can be merged. This will prove that T″ is minimal. Since pushing only affects the output labels, T′ and T″ have the same set of states: Q′=Q″. Hence S_{1 }and S_{2 }are also states of T′. The states of T′ can be viewed as weighted subsets whose state elements belong to T_{1}, because T′ is obtained by determinization of T^{R} _{1}.
Let (q,c) εQ_{1 }x , be a pair of the subset S_{1}. Since T_{1 }is trim there exists ωεΣ* such that δ_{1}(i_{1},ω)=q, so δ′(S_{1}, ω) εF′. Since S_{1 }and S_{2 }are equivalent, we also have: δ′ (S_{2},ω)εF′. Since T_{1 }is subsequential, there exists only one state of T_{1} ^{R }admitting a path labeled with ω to i_{1}, that state is q. Thus, qεS_{2}. Therefore any state q member of a pair of S_{1 }is also member of a pair of S_{2}. By symmetry the reverse is also true. Thus exactly the same states are members of the pairs of S1 and S2. There exists k≧0 such that:
Weights are also the same in S_{1 }and S_{2}. Let Π_{j}, (0≧j≧k), be the set of strings labeling the paths from i_{1 }to q_{i }in T_{1}. σ_{1}(i_{1},ω) is the weight output corresponding to a string ωεΠ_{j}. Consider the accumulated weights c_{ij}, 1≦i≦2,0≦j≦k, in determinization of T^{R} _{1}. Each c_{1j }for instance corresponds to the weight not yet output in the paths reach S_{1}. It needs to be added to the weights of any path from q_{j }εS_{1 }to a final state in rev(T_{1}). In other terms, the determinization algorithm will assign the weight c_{1j}+σ_{1}(i_{1},ω)+λ_{1 }to a path labeled with ω^{R }reaching a final state of T′ from S_{1}. T″ is obtained by pushing from T′. Therefore, the weight of such a path in T″ is:
Similarly, the weight of a path labeled with the same string considered from S_{2 }is:
Since S_{1 }and S_{2 }are equivalent in T″ the weights of the paths from S_{1 }or S_{2 }to F′ are equal. Thus, ∀ωεπ_{j}, ∀jε[0,k],
In the determinization section, the minimum weight of the pairs of any subset is 0. Therefore, ∀jε[0,k], c_{1j}=c_{2j }and S_{2}=S_{1}.
FIGS. 1113 illustrate the minimization of string to weight transducers using the determinization algorithm. The transducer β_{2 }of FIG. 11 is obtained from that of FIG. 7, β_{1}, by reversing it. The application of determinization to β_{2 }results in β_{3 }as shown in FIG. 12. Notice that since β_{1 }is subsequential, according to the theorem the transducer β_{3 }is minimal too. β_{3 }is then reversed and determinized. The resulting transducer β_{4 }in FIG. 13 is minimal and equivalent to β_{1}. One can compare the transducer β_{4 }to the transducer of FIG. 9, δ_{1}. Both are minimal and realize the same function. δ_{1 }provides output weights as soon as possible. It can be obtained from β_{4 }by pushing.
6. Application to Speech Recognition
In the previous sections we gave a theoretical description of the determinization and minimization algorithms for string to weight transducers. Here we indicate their use in practice by describing applications of the algorithms for speech recognition.
String to weight transducers are found at several steps of speech recognition. Phone lattices, language models, and word lattices are typically represented by such transducers. Weights in these graphs correspond to negative logarithms of probabilities. Therefore, they are added along a path. For a given string there might be many different paths in a transducer. The minimum of the total weights of these paths is only considered as relevant information. Thus, the main operations involved in the interpretation of these transducers are addition and min, which are those of the tropical semirings. Thus the algorithms we defined in the previous sections apply to speech recognition.
In the preferred embodiment of the present invention, word lattices are obtained by combining distributions over acoustic observations with acoustic models including context dependency restrictions, combined with pronunciation dictionaries and language models. For a given utterance, the word lattice obtained in such a way contains many paths labeled with the possible sentences and their probabilities. These lattices can be searched to find the most probable sentences which correspond to the best paths, the paths with the smallest weights.
FIG. 14 corresponds to a typical word lattice W_{1 }obtained in speech recognition. It corresponds to the following utterance: “Which flights leave Detroit and arrive at SaintPetersburg around nine a.m.?” Clearly the lattice is complex and contains much redundancy. Many sentences are repeated in the lattice with different weights. It contains about 83 million paths. Usually it is not enough to consider the best path. One needs to correct the best path approximation by considering the n best paths, where the value of n depends on the task considered. Notice that in case n is very large one would need to consider all these 83 million paths. The transducer contains 106 states and 359 transitions. Determinization applies to this lattice. The resulting transducer W_{2 }as shown in FIG. 15 is sparser. Recall that it is equivalent to W_{1}. It realizes exactly the same function mapping strings to weights. For a given sentence s recognized by W_{1 }there are many different paths with different total weights. W_{2 }contains a path labeled with s and with a total weight equal to the minimum of the weights of the paths of W_{1}. No pruning, heuristics or approximation has been used here. The lattice W_{2 }only contains 18 paths. It is not hard to realize that the search stage in speech recognition is greatly simplified when applied to W_{2 }rather than W_{1}. W_{2 }admits 38 states and 51 transitions.
The transducer W_{2 }can still be minimized. The minimization algorithm described in the previous section leads to the transducer W_{3 }as shown in FIG. 16. It contains 25 states and 33 transitions and of course the same number of paths as W_{2}, 18. The effect of minimization appears as less important. This is because determinization includes here a large part of the minimization by reducing the size of the first lattice. This can be explained by the degree of nondeterminism of word lattices such as W_{1}. Many states can be reached by the same set of strings. These states are grouped into a single subset during determinization.
Also the complexity of determinization is exponential in general, but in the case of the lattices considered in speech recognition, this is not the case. A more specific determinization can be used in the cases very often encountered in natural language processing where the graph admits a loop at the initial state over the elements of the alphabet. Since they contain much redundancy the resulting lattice is smaller than the initial. In fact, the time complexity of determinization can be expressed in terms of the initial and resulting lattices WI_{1 }and W_{2 }by OJA log A(W_{1}W_{2})^{2}), where W_{1} and W_{2 }denote the sizes of W_{1 }and W_{2}. Clearly if we restrict determinization to the cases where W_{2}≦W_{1} its complexity is polynomial in terms of the size of the initial transducer W_{1}. The same remark applies to the space complexity of the algorithm. In practice the algorithm appears to be very efficient. As an example, it takes about 0.1s on a Silicon Graphics (indy 100 MHZ Processor, 64 Mb RAM; The main part of this time corresponds to I/O's and is therefore independent of the algorithm.) to determinize the transducer W_{1 }of FIG. 14.
Determinization makes the use of lattices much faster. Since at any state there exists at most one transition labeled with the word considered, finding the weight associated with a sentence does not depend on the size of the lattice. The time and space complexity of such an operation is simply linear in the size of the sentence. When dealing with large tasks, most speech recognition systems use a rescoring method. This consists in first using a poor acoustic or grammar model to produce a word lattice, and then to reuse this word lattice with a more sophisticated model. The size of the word lattice is then a critical parameter in time and space efficient of the system. The determinization and minimization algorithms we presented allow one to considerably reduce the size of such word lattices as seen in the examples that experimented both determinization and minimization algorithms in the ATIS task. The table in FIG. 17 illustrates these results. It shows these algorithms to be very efficient in reducing the redundancy of speech networks in this task. The reduction is also illustrated in the exemplary ATIS task as follows:
Example of word lattice in the ATIS task.
States: 187→37
Transitions: 993→59
Paths: >2^{32}→6993
The number of paths of the word lattice before determinization was larger than that of the largest integer representable with 32 bits machines. We also experimented the minimization algorithm by applying it to several word lattices obtained in the NAB task. These lattices were already detesrminized. The table in FIG. 18 shows the average reduction factors we obtained when using the minimization algorithms with several deterministic lattices obtained for utterances of the NAB task. The reduction factors help to measure the gain of minimization alone since the lattices are already deterministic. The figures of the following example correspond to a typical case. Here is an example of reduction we obtained.
Example of word lattice in NAB task.
Transitions: 108211→37563
States: 10407→2553
An important characteristic of the determinization algorithm is that it can be used on thefly. Indeed, the determinization algorithm is such that given a subset representing state of the resulting transducer, the definition of the transitions leaving that state only depends on that state or equivalently on the underlying subset, and on the transducer to determinize. We have given an implementation of the determinization which allows one both to completely expand the result or to expand it on demand. Arcs leaving state of the determinized transducer are expanded only if needed. This characteristic of the implementation is important. It can then be used for instance at any step in onthefly cascade of composition of transducers in speech recognition to expand only the necessary part of a lattice or transducer. One of the essential implications of the implementation is that it contributes to saving space during the search stage. It is also very useful in speeding up the nbest decoder in speech recognition.
It will be understood that the foregoing is only illustrative of the principles of the invention and that various modifications can be made by those skilled in the art without departing from the scope and spirit of the invention. For example, the determinization and minimization algorithms for string to weight transducers have other applications in speech processing. Many new experiments can be done using these algorithms at different stages of the recognition. They might lead to reshape some of the methods used in this field in particular by providing a new success of the theory of automata and transducers.
We briefly presented the theoretical bases, algorithmic tools and the practical use of set of devices which seem to fit the complexity of language and provide efficiency space and time. It helps to describe the possibilities they offer and to guide algorithmic choices. The notion of determinization can be generalized to that of εdeterminization, for instance, requiring more general algorithms. Other extensions of sequential transducers such as polysubsequential transducers can be considered. New characterizations of rational function in terms of groups could shed new light on some aspects of the theory of finite transducers. The generalization of the operations in the preferred embodiment of the invention is based on that of the notions of semirings and power series.
Sequential transducers admit very efficient algorithms. Sequential machines lead to useful algorithms in many other areas of computational linguistics. In particular, subsequential power series allow one to achieve efficient results in indexation of natural language texts. Furthermore more precision in acoustic modeling, finer language models, fine and large lexicon grammars, a larger vocabulary will lead in the near future to networks of much larger sizes in speech recognition. The determinization and minimization algorithms may limit the size of these networks while keeping their time efficiency. These algorithms also apply to texttospeech synthesis. In fact, the same operations of composition of transducers and perhaps more important size issues can be found in this field. Other specific example of possible uses of the invention include its application in optical character recognition (“OCR”), sampling of genes or biological samples, machine translation (e.g., speechtospeech of texttotext), texttospeech synthesis, handwriting recognition search engine in worldwide web and even stability control of an aircraft.
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Cited By (57)
Publication number  Priority date  Publication date  Assignee  Title 

US20020077833A1 (en) *  20001220  20020620  Arons Barry M.  Transcription and reporting system 
US20020156616A1 (en) *  20010216  20021024  Russell Dale W.  Interactive tool for semiautomatic creation of a natural language grammar from a domain model 
US6505157B1 (en) *  19990301  20030107  Canon Kabushiki Kaisha  Apparatus and method for generating processor usable data from natural language input data 
US20030046055A1 (en) *  20000403  20030306  Xerox Corporation  Method and apparatus for factoring unambiguous finite state transducers 
US20030055641A1 (en) *  20010917  20030320  Yi Jon RongWei  Concatenative speech synthesis using a finitestate transducer 
US20030115169A1 (en) *  20011217  20030619  Hongzhuan Ye  System and method for management of transcribed documents 
US20030120480A1 (en) *  20011115  20030626  Mehryar Mohri  Systems and methods for generating weighted finitestate automata representing grammars 
US6587844B1 (en) *  20000201  20030701  At&T Corp.  System and methods for optimizing networks of weighted unweighted directed graphs 
US20030130843A1 (en) *  20011217  20030710  Ky Dung H.  System and method for speech recognition and transcription 
US20030163452A1 (en) *  20020222  20030828  Chang Jane Wen  Direct navigation for information retrieval 
US20040030556A1 (en) *  19991112  20040212  Bennett Ian M.  Speech based learning/training system using semantic decoding 
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US6714905B1 (en) *  20000502  20040330  Iphrase.Com, Inc.  Parsing ambiguous grammar 
US20040064306A1 (en) *  20020930  20040401  Wolf Peter P.  Voice activated music playback system 
US20040117184A1 (en) *  20021213  20040617  Xerox Corporation  Method and apparatus for recognizing multiword expressions 
US20040133603A1 (en) *  20000502  20040708  Iphrase.Com, Inc., A Delaware Corporation  Prose feedback in information access system 
US20050015734A1 (en) *  20030716  20050120  Microsoft Corporation  Method and apparatus for minimizing weighted networks with link and node labels 
US20050119896A1 (en) *  19991112  20050602  Bennett Ian M.  Adjustable resource based speech recognition system 
US7050977B1 (en)  19991112  20060523  Phoenix Solutions, Inc.  Speechenabled server for internet website and method 
US7136846B2 (en)  20010406  20061114  2005 Keel Company, Inc.  Wireless information retrieval 
US7209880B1 (en) *  20010320  20070424  At&T Corp.  Systems and methods for dynamic reconfigurable speech recognition 
US7219058B1 (en) *  20001013  20070515  At&T Corp.  System and method for processing speech recognition results 
US7240004B1 (en) *  20020304  20070703  At&T Corp.  Systems and methods for determining the determinizability of finitestate automata and transducers 
US20070192104A1 (en) *  20060216  20070816  At&T Corp.  A system and method for providing large vocabulary speech processing based on fixedpoint arithmetic 
US7289948B1 (en)  20020107  20071030  At&T Corp.  Systems and methods for regularly approximating contextfree grammars through transformation 
US20070276664A1 (en) *  20040826  20071129  Khosla Ashok M  Method and system to generate finite state grammars using sample phrases 
US20070288231A1 (en) *  20060608  20071213  Microsoft Corporation Microsoft Patent Group  Uncertainty interval content sensing 
US20080046242A1 (en) *  20000720  20080221  Microsoft Corporation  Compact Easily Parseable Binary Format for a ContextFree Grammar 
US20080120092A1 (en) *  20061120  20080522  Microsoft Corporation  Phrase pair extraction for statistical machine translation 
US7398197B1 (en)  20020107  20080708  At&T Corp.  Systems and methods for generating weighted finitestate automata representing grammars 
US20080208821A1 (en) *  20000502  20080828  International Business Machines Corporation  Information retrieval from a collection of data 
US20090150152A1 (en) *  20071118  20090611  Nice Systems  Method and apparatus for fast search in callcenter monitoring 
US7657424B2 (en)  19991112  20100202  Phoenix Solutions, Inc.  System and method for processing sentence based queries 
US7752159B2 (en)  20010103  20100706  International Business Machines Corporation  System and method for classifying text 
US7756810B2 (en)  20030506  20100713  International Business Machines Corporation  Software tool for training and testing a knowledge base 
US20120078607A1 (en) *  20100929  20120329  Kabushiki Kaisha Toshiba  Speech translation apparatus, method and program 
US8290768B1 (en)  20000621  20121016  International Business Machines Corporation  System and method for determining a set of attributes based on content of communications 
US20130138441A1 (en) *  20111128  20130530  Electronics And Telecommunications Research Institute  Method and system for generating search network for voice recognition 
US20130144604A1 (en) *  20111205  20130606  Infosys Limited  Systems and methods for extracting attributes from text content 
US8478732B1 (en)  20000502  20130702  International Business Machines Corporation  Database aliasing in information access system 
US8543398B1 (en)  20120229  20130924  Google Inc.  Training an automatic speech recognition system using compressed word frequencies 
US8554559B1 (en)  20120713  20131008  Google Inc.  Localized speech recognition with offload 
US8571859B1 (en)  20120531  20131029  Google Inc.  Multistage speaker adaptation 
US20140046651A1 (en) *  20120813  20140213  Xerox Corporation  Solution for maxstring problem and translation and transcription systems using same 
US8805684B1 (en)  20120531  20140812  Google Inc.  Distributed speaker adaptation 
US8943006B2 (en)  20110916  20150127  Kabushiki Kaisha Toshiba  Automaton determinization method, device, and computer program product that involves a plurality of states and deleting states that are next states to first transitions 
US20150106100A1 (en) *  20021031  20150416  Promptu Systems Corporation  Efficient empirical computation and utilization of acoustic confusability 
US20150186756A1 (en) *  20131231  20150702  Google Inc.  Pruning and label selection in hidden markov modelbased ocr 
US20150220074A1 (en) *  20140131  20150806  Kabushiki Kaisha Toshiba  Conversion device, pattern recognition system, conversion method, and computer program product 
US9123333B2 (en)  20120912  20150901  Google Inc.  Minimum bayesian risk methods for automatic speech recognition 
US9202461B2 (en)  20120426  20151201  Google Inc.  Sampling training data for an automatic speech recognition system based on a benchmark classification distribution 
US20160042732A1 (en) *  20050826  20160211  At&T Intellectual Property Ii, L.P.  System and method for robust access and entry to large structured data using voice formfilling 
US20160093292A1 (en) *  20140926  20160331  Intel Corporation  Optimizations to decoding of wfst models for automatic speech recognition 
US20160188573A1 (en) *  20140916  20160630  Voicebox Technologies Corporation  Integration of domain information into state transitions of a finite state transducer for natural language processing 
US9460088B1 (en) *  20130531  20161004  Google Inc.  Writtendomain language modeling with decomposition 
US9584665B2 (en)  20000621  20170228  International Business Machines Corporation  System and method for optimizing timing of responses to customer communications 
US9699129B1 (en)  20000621  20170704  International Business Machines Corporation  System and method for increasing email productivity 
Families Citing this family (12)
Publication number  Priority date  Publication date  Assignee  Title 

US20020116196A1 (en) *  19981112  20020822  Tran Bao Q.  Speech recognizer 
WO2000038175A1 (en) *  19981221  20000629  Koninklijke Philips Electronics N.V.  Language model based on the speech recognition history 
WO2000060577A1 (en) *  19990401  20001012  Koninklijke Philips Electronics N.V.  Multiple stage speech recognizer 
US8260614B1 (en) *  20000928  20120904  Intel Corporation  Method and system for expanding a word graph to a phone graph based on a crossword acoustical model to improve continuous speech recognition 
CA2524199C (en) *  20030429  20130806  Bradford Craig Starkie  A system and process for grammatical inference 
WO2006127504A3 (en) *  20050520  20070628  Sony Computer Entertainment Inc  Optimisation of a grammar for speech recognition 
US8214213B1 (en)  20060427  20120703  At&T Intellectual Property Ii, L.P.  Speech recognition based on pronunciation modeling 
US7902447B1 (en)  20061003  20110308  Sony Computer Entertainment Inc.  Automatic composition of sound sequences using finite state automata 
US20080208566A1 (en) *  20070223  20080828  Microsoft Corporation  Automated wordform transformation and part of speech tag assignment 
US8065300B2 (en) *  20080312  20111122  At&T Intellectual Property Ii, L.P.  Finding the website of a business using the business name 
JP4977163B2 (en)  20090330  20120718  株式会社東芝  Finitestate transducer determining apparatus and finite state transducer determining method 
US8560318B2 (en)  20100514  20131015  Sony Computer Entertainment Inc.  Methods and system for evaluating potential confusion within grammar structure for set of statements to be used in speech recognition during computing event 
Citations (2)
Publication number  Priority date  Publication date  Assignee  Title 

US5495409A (en) *  19930129  19960227  Matsushita Electric Industrial Co., Ltd.  Constructing method of finitestate machine performing transitions according to a partial type of success function and a failure function 
US5594641A (en) *  19920720  19970114  Xerox Corporation  Finitestate transduction of related word forms for text indexing and retrieval 
Family Cites Families (4)
Publication number  Priority date  Publication date  Assignee  Title 

GB8906558D0 (en) *  19890322  19890504  British Telecomm  Pattern recognition 
JP3599775B2 (en) *  19930421  20041208  ゼロックス コーポレイションＸｅｒｏｘ Ｃｏｒｐｏｒａｔｉｏｎ  Finitestate coding system on hyphenation rules 
JP2704106B2 (en) *  19931221  19980126  宮城日本電気株式会社  Data signal multiplexing transmission apparatus for frame synchronization method 
JPH08129603A (en) *  19941101  19960521  Hitachi Ltd  Optical character reader 
Patent Citations (2)
Publication number  Priority date  Publication date  Assignee  Title 

US5594641A (en) *  19920720  19970114  Xerox Corporation  Finitestate transduction of related word forms for text indexing and retrieval 
US5495409A (en) *  19930129  19960227  Matsushita Electric Industrial Co., Ltd.  Constructing method of finitestate machine performing transitions according to a partial type of success function and a failure function 
NonPatent Citations (3)
Title 

John R. Deller et al., "DiscreteTime Processing of Speech Signals", Prentice Hall, 1993 pp. 754755 and 754. 
Mohri "On Some Applications of FiniteState Automata Theory to Natural Language Processing" May 9, 1995, pp. 117.* 
Wayne Wolf, "An Algorithm for NearlyMinimal Collapsing of FiniteState Machine Networks", IEEE Comput. Soc. Press, Nov. 15, 1990, pp. 8083. 
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US7225125B2 (en)  19991112  20070529  Phoenix Solutions, Inc.  Speech recognition system trained with regional speech characteristics 
US7647225B2 (en)  19991112  20100112  Phoenix Solutions, Inc.  Adjustable resource based speech recognition system 
US7657424B2 (en)  19991112  20100202  Phoenix Solutions, Inc.  System and method for processing sentence based queries 
US7725307B2 (en)  19991112  20100525  Phoenix Solutions, Inc.  Query engine for processing voice based queries including semantic decoding 
US7672841B2 (en)  19991112  20100302  Phoenix Solutions, Inc.  Method for processing speech data for a distributed recognition system 
US9190063B2 (en)  19991112  20151117  Nuance Communications, Inc.  Multilanguage speech recognition system 
US7698131B2 (en)  19991112  20100413  Phoenix Solutions, Inc.  Speech recognition system for client devices having differing computing capabilities 
US7702508B2 (en)  19991112  20100420  Phoenix Solutions, Inc.  System and method for natural language processing of query answers 
US9076448B2 (en)  19991112  20150707  Nuance Communications, Inc.  Distributed real time speech recognition system 
US20040030556A1 (en) *  19991112  20040212  Bennett Ian M.  Speech based learning/training system using semantic decoding 
US7725321B2 (en)  19991112  20100525  Phoenix Solutions, Inc.  Speech based query system using semantic decoding 
US7729904B2 (en)  19991112  20100601  Phoenix Solutions, Inc.  Partial speech processing device and method for use in distributed systems 
US8762152B2 (en)  19991112  20140624  Nuance Communications, Inc.  Speech recognition system interactive agent 
US7277854B2 (en)  19991112  20071002  Phoenix Solutions, Inc  Speech recognition system interactive agent 
US7203646B2 (en)  19991112  20070410  Phoenix Solutions, Inc.  Distributed internet based speech recognition system with natural language support 
US7831426B2 (en)  19991112  20101109  Phoenix Solutions, Inc.  Network based interactive speech recognition system 
US8352277B2 (en)  19991112  20130108  Phoenix Solutions, Inc.  Method of interacting through speech with a webconnected server 
US20050119896A1 (en) *  19991112  20050602  Bennett Ian M.  Adjustable resource based speech recognition system 
US7873519B2 (en)  19991112  20110118  Phoenix Solutions, Inc.  Natural language speech lattice containing semantic variants 
US7912702B2 (en)  19991112  20110322  Phoenix Solutions, Inc.  Statistical language model trained with semantic variants 
US7139714B2 (en)  19991112  20061121  Phoenix Solutions, Inc.  Adjustable resource based speech recognition system 
US8229734B2 (en)  19991112  20120724  Phoenix Solutions, Inc.  Semantic decoding of user queries 
US7050977B1 (en)  19991112  20060523  Phoenix Solutions, Inc.  Speechenabled server for internet website and method 
US20060200353A1 (en) *  19991112  20060907  Bennett Ian M  Distributed Internet Based Speech Recognition System With Natural Language Support 
US20060235696A1 (en) *  19991112  20061019  Bennett Ian M  Network based interactive speech recognition system 
US7725320B2 (en)  19991112  20100525  Phoenix Solutions, Inc.  Internet based speech recognition system with dynamic grammars 
US6587844B1 (en) *  20000201  20030701  At&T Corp.  System and methods for optimizing networks of weighted unweighted directed graphs 
US6944588B2 (en) *  20000403  20050913  Xerox Corporation  Method and apparatus for factoring unambiguous finite state transducers 
US20030046055A1 (en) *  20000403  20030306  Xerox Corporation  Method and apparatus for factoring unambiguous finite state transducers 
US7702677B2 (en)  20000502  20100420  International Business Machines Corporation  Information retrieval from a collection of data 
US20080208821A1 (en) *  20000502  20080828  International Business Machines Corporation  Information retrieval from a collection of data 
US20040133603A1 (en) *  20000502  20040708  Iphrase.Com, Inc., A Delaware Corporation  Prose feedback in information access system 
US6714905B1 (en) *  20000502  20040330  Iphrase.Com, Inc.  Parsing ambiguous grammar 
US8645405B2 (en)  20000502  20140204  International Business Machines Corporation  Natural language expression in response to a query 
US7243095B2 (en)  20000502  20070710  International Business Machines Corporation  Prose feedback in information access system 
US8478732B1 (en)  20000502  20130702  International Business Machines Corporation  Database aliasing in information access system 
US9699129B1 (en)  20000621  20170704  International Business Machines Corporation  System and method for increasing email productivity 
US8290768B1 (en)  20000621  20121016  International Business Machines Corporation  System and method for determining a set of attributes based on content of communications 
US9584665B2 (en)  20000621  20170228  International Business Machines Corporation  System and method for optimizing timing of responses to customer communications 
US20080046242A1 (en) *  20000720  20080221  Microsoft Corporation  Compact Easily Parseable Binary Format for a ContextFree Grammar 
US7571096B2 (en) *  20000720  20090804  Microsoft Corporation  Speech recognition using a stateandtransition based binary speech grammar with a last transition value 
US7219058B1 (en) *  20001013  20070515  At&T Corp.  System and method for processing speech recognition results 
US8346550B2 (en)  20001013  20130101  At&T Intellectual Property Ii, L.P.  System and method for processing speech recognition 
US7904294B1 (en)  20001013  20110308  At&T Intellectual Property Ii, L.P.  System and method for processing speech recognition 
US8571861B2 (en)  20001013  20131029  At&T Intellectual Property Ii, L.P.  System and method for processing speech recognition 
US20110137651A1 (en) *  20001013  20110609  At&T Intellectual Property Ii, L.P.  System and Method for Processing Speech Recognition 
US20020077833A1 (en) *  20001220  20020620  Arons Barry M.  Transcription and reporting system 
US7752159B2 (en)  20010103  20100706  International Business Machines Corporation  System and method for classifying text 
US20020156616A1 (en) *  20010216  20021024  Russell Dale W.  Interactive tool for semiautomatic creation of a natural language grammar from a domain model 
US20090006088A1 (en) *  20010320  20090101  At&T Corp.  System and method of performing speech recognition based on a user identifier 
US7209880B1 (en) *  20010320  20070424  At&T Corp.  Systems and methods for dynamic reconfigurable speech recognition 
US7451081B1 (en)  20010320  20081111  At&T Corp.  System and method of performing speech recognition based on a user identifier 
US9058810B2 (en)  20010320  20150616  At&T Intellectual Property Ii, L.P.  System and method of performing userspecific automatic speech recognition 
US8145481B2 (en)  20010320  20120327  At&T Intellectual Property Ii, L.P.  System and method of performing userspecific automatic speech recognition 
US7136846B2 (en)  20010406  20061114  2005 Keel Company, Inc.  Wireless information retrieval 
US20030055641A1 (en) *  20010917  20030320  Yi Jon RongWei  Concatenative speech synthesis using a finitestate transducer 
US7165030B2 (en) *  20010917  20070116  Massachusetts Institute Of Technology  Concatenative speech synthesis using a finitestate transducer 
US20030120480A1 (en) *  20011115  20030626  Mehryar Mohri  Systems and methods for generating weighted finitestate automata representing grammars 
US7181386B2 (en)  20011115  20070220  At&T Corp.  Systems and methods for generating weighted finitestate automata representing grammars 
US6990445B2 (en)  20011217  20060124  Xl8 Systems, Inc.  System and method for speech recognition and transcription 
US20030130843A1 (en) *  20011217  20030710  Ky Dung H.  System and method for speech recognition and transcription 
US20030115169A1 (en) *  20011217  20030619  Hongzhuan Ye  System and method for management of transcribed documents 
US8543383B2 (en)  20020107  20130924  At&T Intellectual Property Ii, L.P.  Systems and methods for generating weighted finitestate automata representing grammars 
US7716041B2 (en)  20020107  20100511  At&T Intellectual Property Ii, L.P.  Systems and methods for regularly approximating contextfree grammars through transformation 
US20080010059A1 (en) *  20020107  20080110  At & T Corp.  Systems and methods for regularly approximating contextfree grammars through transformation 
US20080243484A1 (en) *  20020107  20081002  At&T Corp.  Systems and methods for generating weighted finitestate automata representing grammars 
US7289948B1 (en)  20020107  20071030  At&T Corp.  Systems and methods for regularly approximating contextfree grammars through transformation 
US8050908B2 (en)  20020107  20111101  At&T Intellectual Property Ii, L.P.  Systems and methods for generating weighted finitestate automata representing grammars 
US7398197B1 (en)  20020107  20080708  At&T Corp.  Systems and methods for generating weighted finitestate automata representing grammars 
US7783643B2 (en)  20020222  20100824  International Business Machines Corporation  Direct navigation for information retrieval 
US20030163452A1 (en) *  20020222  20030828  Chang Jane Wen  Direct navigation for information retrieval 
US7343372B2 (en)  20020222  20080311  International Business Machines Corporation  Direct navigation for information retrieval 
US20080140613A1 (en) *  20020222  20080612  International Business Machines Corporation  Direct navigation for information retrieval 
US7783485B2 (en)  20020304  20100824  At&T Intellectual Property Ii, L.P.  Systems and methods for determining the determinizability of finitestate automata and transducers 
US20070299668A1 (en) *  20020304  20071227  Cyril Allauzen  Systems and methods for determining the determinizability of finitestate automata and transducers 
US7240004B1 (en) *  20020304  20070703  At&T Corp.  Systems and methods for determining the determinizability of finitestate automata and transducers 
US8234115B2 (en)  20020329  20120731  At&T Intellectual Property Ii, L.P.  Systems and methods for determining the Nbest strings 
EP1398759A3 (en) *  20020329  20041229  AT&T Corp.  Systems and methods for determining the Nbest strings of an automaton 
EP1398759A2 (en) *  20020329  20040317  AT&T Corp.  Systems and methods for determining the Nbest strings of an automaton 
US8527273B2 (en) *  20020329  20130903  At&T Intellectual Property Ii, L.P.  Systems and methods for determining the Nbest strings 
US20040064306A1 (en) *  20020930  20040401  Wolf Peter P.  Voice activated music playback system 
US20150106100A1 (en) *  20021031  20150416  Promptu Systems Corporation  Efficient empirical computation and utilization of acoustic confusability 
US9626965B2 (en) *  20021031  20170418  Promptu Systems Corporation  Efficient empirical computation and utilization of acoustic confusability 
US20040117184A1 (en) *  20021213  20040617  Xerox Corporation  Method and apparatus for recognizing multiword expressions 
US7346511B2 (en) *  20021213  20080318  Xerox Corporation  Method and apparatus for recognizing multiword expressions 
US8495002B2 (en)  20030506  20130723  International Business Machines Corporation  Software tool for training and testing a knowledge base 
US7756810B2 (en)  20030506  20100713  International Business Machines Corporation  Software tool for training and testing a knowledge base 
US20050015734A1 (en) *  20030716  20050120  Microsoft Corporation  Method and apparatus for minimizing weighted networks with link and node labels 
US20050149888A1 (en) *  20030716  20050707  Microsoft Corporation  Method and apparatus for minimizing weighted networks with link and node labels 
US7401303B2 (en)  20030716  20080715  Microsoft Corporation  Method and apparatus for minimizing weighted networks with link and node labels 
US7003740B2 (en) *  20030716  20060221  Microsoft Corporation  Method and apparatus for minimizing weighted networks with link and node labels 
US8396712B2 (en) *  20040826  20130312  West Corporation  Method and system to generate finite state grammars using sample phrases 
US8712775B1 (en) *  20040826  20140429  West Interactive Corporation Ii  Method and system to generate finite state grammars using sample phrases 
US20070276664A1 (en) *  20040826  20071129  Khosla Ashok M  Method and system to generate finite state grammars using sample phrases 
US9824682B2 (en) *  20050826  20171121  Nuance Communications, Inc.  System and method for robust access and entry to large structured data using voice formfilling 
US20160042732A1 (en) *  20050826  20160211  At&T Intellectual Property Ii, L.P.  System and method for robust access and entry to large structured data using voice formfilling 
US20070192104A1 (en) *  20060216  20070816  At&T Corp.  A system and method for providing large vocabulary speech processing based on fixedpoint arithmetic 
US8195462B2 (en) *  20060216  20120605  At&T Intellectual Property Ii, L.P.  System and method for providing large vocabulary speech processing based on fixedpoint arithmetic 
US20070288231A1 (en) *  20060608  20071213  Microsoft Corporation Microsoft Patent Group  Uncertainty interval content sensing 
US8209175B2 (en) *  20060608  20120626  Microsoft Corporation  Uncertainty interval content sensing within communications 
US20080120092A1 (en) *  20061120  20080522  Microsoft Corporation  Phrase pair extraction for statistical machine translation 
US7788095B2 (en) *  20071118  20100831  Nice Systems, Ltd.  Method and apparatus for fast search in callcenter monitoring 
US20090150152A1 (en) *  20071118  20090611  Nice Systems  Method and apparatus for fast search in callcenter monitoring 
US8635070B2 (en) *  20100929  20140121  Kabushiki Kaisha Toshiba  Speech translation apparatus, method and program that generates insertion sentence explaining recognized emotion types 
US20120078607A1 (en) *  20100929  20120329  Kabushiki Kaisha Toshiba  Speech translation apparatus, method and program 
US8943006B2 (en)  20110916  20150127  Kabushiki Kaisha Toshiba  Automaton determinization method, device, and computer program product that involves a plurality of states and deleting states that are next states to first transitions 
US9286891B2 (en)  20110916  20160315  Kabushiki Kaisha Toshiba  Automaton determinization method, device, and computer program product involving deleting states 
US20130138441A1 (en) *  20111128  20130530  Electronics And Telecommunications Research Institute  Method and system for generating search network for voice recognition 
US20130144604A1 (en) *  20111205  20130606  Infosys Limited  Systems and methods for extracting attributes from text content 
US8543398B1 (en)  20120229  20130924  Google Inc.  Training an automatic speech recognition system using compressed word frequencies 
US9202461B2 (en)  20120426  20151201  Google Inc.  Sampling training data for an automatic speech recognition system based on a benchmark classification distribution 
US8571859B1 (en)  20120531  20131029  Google Inc.  Multistage speaker adaptation 
US8805684B1 (en)  20120531  20140812  Google Inc.  Distributed speaker adaptation 
US8880398B1 (en)  20120713  20141104  Google Inc.  Localized speech recognition with offload 
US8554559B1 (en)  20120713  20131008  Google Inc.  Localized speech recognition with offload 
US20140046651A1 (en) *  20120813  20140213  Xerox Corporation  Solution for maxstring problem and translation and transcription systems using same 
US9123333B2 (en)  20120912  20150901  Google Inc.  Minimum bayesian risk methods for automatic speech recognition 
US9460088B1 (en) *  20130531  20161004  Google Inc.  Writtendomain language modeling with decomposition 
US20160070992A1 (en) *  20131231  20160310  Google Inc.  Pruning and label selection in hidden markov modelbased ocr 
US9189708B2 (en) *  20131231  20151117  Google Inc.  Pruning and label selection in hidden markov modelbased OCR 
US9672448B2 (en) *  20131231  20170606  Google Inc.  Pruning and label selection in Hidden Markov Modelbased OCR 
US20150186756A1 (en) *  20131231  20150702  Google Inc.  Pruning and label selection in hidden markov modelbased ocr 
US20150220074A1 (en) *  20140131  20150806  Kabushiki Kaisha Toshiba  Conversion device, pattern recognition system, conversion method, and computer program product 
US20160188573A1 (en) *  20140916  20160630  Voicebox Technologies Corporation  Integration of domain information into state transitions of a finite state transducer for natural language processing 
US20160093292A1 (en) *  20140926  20160331  Intel Corporation  Optimizations to decoding of wfst models for automatic speech recognition 
US9672810B2 (en) *  20140926  20170606  Intel Corporation  Optimizations to decoding of WFST models for automatic speech recognition 
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